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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Assume a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=A x^2+B x+C.
\end{equation*}
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<p class="continuation">Substituting it into the ODE, one has</p>
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\begin{equation*}
\begin{aligned}
&amp;2 A-2 (2 A x+ B)-3 (A x^2 + B x+ C)= 2 x^2 \\
&amp;\to -3 A x^2+(-4 A-3 B)x+2 A-2 B- 3C=2 x^2\\
&amp;\to -3 A=2, \quad -4A-3B=0, \quad 2 A-2 B- 3 C=0\\
&amp;\to A=-\frac{2}{3}, \quad B=\frac{8}{9}, \quad C=-\frac{28}{27}.
\end{aligned}
\end{equation*}
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<p class="continuation">Then the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=-\frac{2}{3} x^2+\frac{8}{9} x-\frac{28}{27}.
\end{equation*}
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<span class="incontext"><a href="sec3_6.html#p-124" class="internal">in-context</a></span>
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